Average Error: 34.0 → 34.1
Time: 39.8s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot {\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{5}\right)\right)\right)}\right)}^{\left(\sqrt[3]{5}\right)}\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot {\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{5}\right)\right)\right)}\right)}^{\left(\sqrt[3]{5}\right)}
double f(double a, double c) {
        double r15120 = c;
        double r15121 = cosh(r15120);
        double r15122 = a;
        double r15123 = log1p(r15122);
        double r15124 = fmod(r15121, r15123);
        return r15124;
}
↓
double f(double a, double c) {
        double r15125 = c;
        double r15126 = cosh(r15125);
        double r15127 = a;
        double r15128 = log1p(r15127);
        double r15129 = fmod(r15126, r15128);
        double r15130 = sqrt(r15129);
        double r15131 = cbrt(r15130);
        double r15132 = 5.0;
        double r15133 = cbrt(r15132);
        double r15134 = log1p(r15133);
        double r15135 = expm1(r15134);
        double r15136 = r15133 * r15135;
        double r15137 = pow(r15131, r15136);
        double r15138 = pow(r15137, r15133);
        double r15139 = r15131 * r15138;
        return r15139;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.0

    \[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt34.0

    \[\leadsto \color{blue}{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt34.0

    \[\leadsto \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}\]

  6. Applied associate-*r*34.0

    \[\leadsto \color{blue}{\left(\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}}\]

  7. Simplified34.1

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(4 + 1\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt34.1

    \[\leadsto {\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\color{blue}{\left(\left(\sqrt[3]{4 + 1} \cdot \sqrt[3]{4 + 1}\right) \cdot \sqrt[3]{4 + 1}\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]

  10. Applied pow-unpow34.1

    \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{4 + 1} \cdot \sqrt[3]{4 + 1}\right)}\right)}^{\left(\sqrt[3]{4 + 1}\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]

  11. Simplified34.1

    \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{5} \cdot \sqrt[3]{5}\right)}\right)}}^{\left(\sqrt[3]{4 + 1}\right)} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]
  12. Using strategy rm

  13. Applied expm1-log1p-u34.1

    \[\leadsto {\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{5}\right)\right)}\right)}\right)}^{\left(\sqrt[3]{4 + 1}\right)} \cdot \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]

  14. Final simplification34.1

    \[\leadsto \sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot {\left({\left(\sqrt[3]{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)}^{\left(\sqrt[3]{5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{5}\right)\right)\right)}\right)}^{\left(\sqrt[3]{5}\right)}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  (fmod (cosh c) (log1p a)))